What is wrong with this "proof" that there is no $\omega$th inaccessible cardinal?

In many of these fake proofs, if you look closely you'll see that the infinity part plays a red herring.

The same argument, supposedly, would have been that if there are two inaccessible cardinals, $\kappa<\lambda$, then $V_\lambda$ would satisfy "There exists an inaccessible cardinal", thus the theory "$\sf ZFC$+There exists an inaccessible cardinal" proves its own consistency.

But all that we did was to show that there is one model of $\sf ZFC$ with an inaccessible; and in fact we just showed that $\sf ZFC$ with two inaccessible cardinals prove the consistency of $\sf ZFC$ with one inaccessible; while $V_\kappa$ is a model of $\sf ZFC$.

And of course the mistake is easily noted here, one inaccessible is not the same as two inaccessible cardinals. And we used the fact that there are two inaccessible cardinals and not the one.

The same mistake, as noted in the comments and the edit, was made here. You have assumed that $\omega$ inaccessibles mean $\omega+1$ inaccessibles. But there is a very large gap between $\sup\kappa_n$ and $\kappa_\omega$.

More to the point, you should measure order type and "discernible properties". So you shouldn't say "There are $\aleph_0$ inaccessible cardinals" but rather "There are $\omega$ inaccessible cardinals" vs. "There are $\omega+1$ inaccessible cardinals".

This can blow up into undefinable order types, or into proper classes, and indeed at some point you will get some large cardinal $\lambda$ such that $V_\kappa$ is an elementary model of $V_\lambda$ for an inaccessible cardinal $\kappa$. In particular all the first-order properties of $\lambda$ will be reflected by $\kappa$. But this just means that we are using properties of $\lambda$ which are first-order expressible in $V$, but not in $V_\lambda$ (as statement about the class of ordinals there).

I don't see the problem with your proof, but I look at it like this. Let $\theta_0$ be the first (I assume always strongly) inaccessible cardinal. Let $\theta_1$ be the smallest inaccessible larger than $\theta_0$, $\theta_2$ be the smallest inaccessible cardinal larger than $\theta_1$ and so forth. So what is $\theta_\omega$? Method 1 says it is the sup or limit of all of the $\theta_i$ as $i$ ranges over the finite ordinals. Method 2 says it is the smallest inaccessible cardinal larger than all the $\theta_i$. These two are NOT the same. The cardinal defined by Method 1 is the limit of a countable set, namely $\theta_i$, so it is accessible; in fact it is singular. The next inaccessible cardinal is therefore $\theta_{\omega+1}$. So it appears that a skip appears every time we hit a subscript that is a limit ordinal; in particular, the set of all inaccessibles less than a large cardinal $\kappa$ is not a club set.